On 23 Sep 2013, at 12:41, Telmo Menezes wrote:

On Sat, Sep 21, 2013 at 9:43 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:

On 21 Sep 2013, at 15:10, Telmo Menezes wrote:

On Fri, Sep 20, 2013 at 3:58 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:


On 19 Sep 2013, at 16:51, Telmo Menezes wrote:

On Thu, Sep 19, 2013 at 4:31 PM, Bruno Marchal <marc...@ulb.ac.be>


If so, can't we say ~D~t and thus []t?


Yes, []t is a theorem, of G and most modal logic, but not of Z!




Isn't the only situation where ~Dt the one where this is no world?


~Dt, that is [] f, inconsistency, is the type of the error, dream, lie, and
"near-death", or in-a-cul-de-sac.

Thus your interest in near-death experiences?


Yes. And in all "extreme" altered state of consciousness. Those extreme cases provide key information.




We should *try* to avoid it, but we can't avoid it without loosing our
universality.

The consistent machines face the dilemma between security and lack of
freedom-universality. With <>p = ~[] ~p, here are equivalent way to write
it:

<>t -> ~[]<>t
<>t -> <> [] f
[]<>t -> [] f

I don't understand how you arrive at this equivalence.

I use only the fact that (p -> q) is equivalent with (~q -> ~p) (the contraposition rule, which is valid in classical propositional logic), and the definition of <> p = ~[] ~p. I use also that ~~p is equivalent with p.

Note that []p = ~~[]~~p = ~<> ~p.  And,

~[]p = <> ~p
and
~<>p = [] ~p

Like with the quantifier, a not (~) jumping above a modal sign makes it into a diamond, if it was a bo, and a box, if it was a diamond.


Starting from <>t -> ~[]<>t. Contraposition gives ~~[]<>t -> ~<>t, and this gives by above, []<>t -> []~t, which gives
[]<>t -> []f   (as ~t = f, and ~f = t).

OK?

For the third one, starting from the first one again: <>t -> ~[]<>t, By contraposition []<>t -> ~<>t , but ~<>t = []~t = [] f.

OK?

Bruno



http://iridia.ulb.ac.be/~marchal/



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